If `f(x)=[sin^(2) x]` ([.] denotes the greatest integer function), then

Options

A. f is everywhere continuous
B. f is everywhere differerntiable
C. f is a constant function
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = [\sin^2 x] \), where \([.]\) denotes the greatest integer function.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) = [\sin^2 x] \) takes the value of \( \sin^2 x \) and applies the greatest integer function to it. The sine function oscillates between -1 and 1, so \( \sin^2 x \) will oscillate between 0 and 1.

2. **Range of \( \sin^2 x \)**:
The maximum value of \( \sin^2 x \) is 1 (when \( \sin x = 1 \)), and the minimum value is 0 (when \( \sin x = 0 \)). Therefore, the range of \( \sin^2 x \) is [0, 1].

3. **Applying the Greatest Integer Function**:
- For \( 0 \leq \sin^2 x < 1 \), \( [\sin^2 x] = 0 \).
- For \( \sin^2 x = 1 \), \( [\sin^2 x] = 1 \).

The value of \( f(x) \) will be:
- \( f(x) = 0 \) for all \( x \) except at points where \( \sin^2 x = 1 \) (which occurs at odd multiples of \( \frac{\pi}{2} \)).
- \( f(x) = 1 \) at \( x = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \).

4. **Identifying Points of Discontinuity**:
The function \( f(x) \) is discontinuous at points where \( \sin^2 x = 1 \) (i.e., \( x = \frac{\pi}{2} + n\pi \)). At these points:
- The left-hand limit and right-hand limit of \( f(x) \) are both 0.
- The value of \( f(x) \) at these points is 1.
Therefore, \( f(x) \) is discontinuous at these points.

5. **Continuity and Differentiability**:
- Since \( f(x) \) is discontinuous at infinitely many points, it cannot be continuous everywhere.
- A function must be continuous at a point to be differentiable there. Therefore, \( f(x) \) is not differentiable anywhere.

6. **Conclusion**:
- \( f(x) \) is not everywhere continuous (Option A is false).
- \( f(x) \) is not everywhere differentiable (Option B is false).
- \( f(x) \) is not a constant function (Option C is false).
- Thus, the correct answer is Option D: None of these.